2.5 Transmissibility Functions

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Transmissibility functions are transfer functions that are particularly useful in the analysis of vibration

isolation in machinery and other mechanical systems. Two types of transmissibility function, force

transmissibility and motion transmissibility, can be defined (see Table 2.2). Due to a reciprocity

characteristic in linear systems, it can be shown that these two transfer functions are equal and,

consequently, it is sufficient to consider only one of them. Let us, however, consider both types first and

show their equivalence.

2.5.1 Force Transmissibility

Consider a mechanical system that is supported on a rigid foundation through a suspension system. If a

forcing excitation is applied to the system, the force is not directly transmitted to the foundation. The

suspension system acts as a vibration isolation device. Force transmissibility determines the fraction of

the forcing excitation that is transmitted to the support structure (foundation) through the suspension,

at different excitation frequencies, and is defined as

Force Transmissibility; Tf ¼

Force Transmitted to Support; Fs

Applied Force; F ð2:91Þ

Note that this is defined in the frequency domain, and that accordingly Fs and F should be interpreted

as the Fourier spectra of the corresponding forces.

A schematic representation of the force transmissibility mechanism is shown in Figure 2.11(a). The

reason for the suspension force fs being not equal to the applied force f is attributed to the inertia paths

(broken line in the figure) that are present in a mechanical system.

2.5.2 Motion Transmissibility

Consider a mechanical system that is supported by suspension on a structure that may be subjected to

undesirable motions (e.g., guideway deflections, vehicle motions, seismic disturbances). Motion

transmissibility determines the fraction of the support motion that is transmitted to the system through

Frequency-Domain Analysis 2-31

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its suspension at different frequencies. It is defined as

Motion Transmissibility; Tm ¼

System Motion; Vm

Support Motion; V ð2:92Þ

The velocities Vm and V are expressed in the frequency domain, as Fourier spectra.

A schematic representation of the motion transmissibility mechanism is shown in Figure 2.11(b).

Typically, the motion of the system is taken as the velocity of one of its critical masses. Different

transmissibility functions are obtained when different mass points (or DoFs) of the system are

considered.

Next, two examples are given to show a reciprocity property that makes the force transmissibility and

the motion transmissibility functions identical in complementary (reciprocal) systems.

System Suspended on a Rigid Base (Force Transmissibility)

Consider the system suspended on a rigid base and excited by force f ðtÞ; as shown earlier in Figure 2.9(a).

Here the system is the inertia element m, and the suspension is the parallel spring and damper

combination. We have noted that the three elements m; k; and b are all in parallel.

Here, Zm ¼ mjv; Zb ¼ b; and Zk ¼ k=jv; as given in Table 2.3. Now, because the elements are corrected

in parallel, we have (see Table 2.4)

f

v ¼ Zm þ Zb þ Zk ð2:93Þ

Hence,

v

f ¼

1

Zm þ Zb þ Zd ð2:94Þ

Also, suspension impedance is

fs

v ¼ Zb þ Zk ¼ Zs ð2:95Þ

where fs is the force transmitted to the support structure (foundation). Then,

fs

f ¼

Zb þ Zk

Zm þ Zb þ Zk ¼

Zs

Zm þ Zs ð2:96Þ

v (t)

vm

Mechanical

System

Suspension

Mechanical

System

Suspension

f (t)

fs

Zs

Zm

v

vm

f

fs =

Forcing

Excitation

Zm

Mm

= 1

Zs

Ms

1 =

(a) (b)

FIGURE 2.11 (a) An inertial system with ground-based suspension; (b) the counterpart (complementary) system

with support motion.

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© 2005 by Taylor & Francis Group, LLC

This result should be immediately clear since force is divided among parallel branches in proportion to

their impedance values (because the velocity is common). Now,

Force Transmissibility Magnitude; lTf l ¼

Force Transmitted to Support

Applied Force to System

􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈

¼

Zs

Zm þ Zs

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

ð2:97Þ

Substitute parameters

lTf l ¼

fs

f

􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈

¼

b þ

k

jv

mjv þ b þ

k

jv

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

¼

bjv þ k

k 2 mv2 þ bjv

􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b2v2 þ k2

ðk 2 mv2Þ2 þ b2v2

s

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð2zvnvÞ2 þ v4

n

ðv2

n 2 v2Þ2 þ ð2zvnvÞ2

s

On simplification, we obtain

lTf l ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 4z2r2

ð1 2 r2Þ2 þ 4z2r2

s

ð2:98Þ

where the normalized frequency is

v

vn ¼ r

At r ¼ 0; we have lTf l ¼ 1.

At r ¼ 1;

lTf l ¼

ffiffiffiffiffiffiffiffiffiffiffi

1 þ

1

4z2

s

ð2:99Þ

This transmissibility magnitude curve is shown in Figure 2.12.

System with Support Motion (Motion Transmissibility)

Consider again the system suspended on a moving platform as shown in Figure 2.10(a). For this system,

we have Zm ¼ mjv and Mm ¼ 1=mjv for the mass element. Since the damper and the spring are

Frequency

Ratio

r

Transmissibility

T

1

0

4z 2

1+ 1

1

FIGURE 2.12 Transmissibility curve for a simple mechanical oscillator.

Frequency-Domain Analysis 2-33

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connected in parallel, the corresponding impedances are additive. Hence, we have

Zs ¼ Zb þ Zk ¼ b þ

k

jv

and Ms ¼

1

b þ

k

jv

Motion Transmissibility Magnitude; Tm ¼

Motion of system inertia

Applied support motion

􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈

¼

Mm

Mm þ Ms

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

ð2:100Þ

This directly follows from the fact that the velocity is divided among series elements in proportion to

their mobilities (because the force is common).

However,

Mm

Mm þ Ms ¼

1

1 þ

Ms

Ms

¼

1

1 þ

Zm

Zs

¼

Zs

Zm þ Zs ð2:101Þ

Hence,

lTml ¼

Zs

Zm þ Zs

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

ð2:102Þ

It follows that

Tf ¼ Tm ð2:103Þ

This establishes the reciprocity property.

2.5.3 General Case

Consider an inertial system with a ground-based suspension, as shown in Figure 2.11(a), and its

counterpart with a moving support, as shown in Figure 2.11(b).

The corresponding impedance circuits are shown in Figure 2.13.

For system (a), we have

fs

f ¼

Zs

Zm þ Zs

For system (b), we have

Vm

v ¼

Mm

Mm þ Ms ¼

1

1 þ

Ms

Mm

¼

1

1 þ

Zm

Zs

¼

Zs

Zs þ Zm

Zs, Ms

v(t)

+

−

vm

Zm, Mm

fs

Zm Zs f(t)

(a) (b)

FIGURE 2.13 Impedance circuits: (a) inertial system with ground-based suspension; (b) system and suspension

with support motion.

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It follows that

fs

f ¼

Vm

v

and Tf ¼ Tm ð2:104Þ

for this more general situation.

Example 2.7

Consider again the problem of a fluid coupling system shown in Figure 2.5 and studied in Example 2.1. A

schematic mechanical circuit for the system is shown in Figure 2.14(a). The corresponding impedance

circuit is shown in Figure 2.14(b). Let us use the impedance method to solve the same problem.

For this problem, we have: input angular velocity ¼ a_ðtÞ; angular velocity of the load ¼ u_; and load

impedance Zl ¼ Zm þ Zk: In view of the series connection of Zb and Zl, we have

u_

a_ ¼

Ml

Ml þ Mb ¼

1=Mb

1=Mb þ 1=Ml ¼

Zb

Zb þ Zm þ Zk ðiÞ

For the torsional spring,

t

u_ ¼ Zk ðiiÞ

Multiply Equation (i) and Equation (ii) together.

t

a_ ¼

ZkZb

ðZb þ Zm þ ZkÞ ðiiiÞ

Since the time derivative corresponds to multiplication by jv in the frequency domain, we can write

(iii) in the form

t

a ¼ jv

ZkZb

ðZb þ Zm þ ZkÞ ðivÞ

Substitute Zb ¼ b; Zm ¼ jvJ; and Zk ¼ k=jv: We obtain t=a ¼ kbjv=ððk 2 Jv2Þ þ bjvÞ; which is

identical to what we obtained in Example 2.1.

2.5.4 Peak Values of Frequency-Response Functions

The peak values of a frequency transfer function correspond to the resonances. The frequencies at these

points are called resonant frequencies. Since a transfer function is the ratio of a response variable to an

input variable, it is reasonable to obtain different peak frequencies for the same excitation input, if the

response variable that is considered is different. Some results obtained for a damped oscillator model are

summarized in Table 2.5.

α (t)

J

+

k b

0

θ

Zm Zk

Zb

θ

α(t)

+

Velocity

Source

(a) (b)

.

.

.

.

FIGURE 2.14 (a) Schematic mechanical circuit of the fluid coupling system; (b) impedance circuit.

Frequency-Domain Analysis 2-35

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TABLE 2.5 Some Practical Frequency-Response Functions and Their Peaks

System Response/Excitation Frequency-Response Function

(Normalized)

Normalized Frequency ðrpÞ Peak Magnitude

(Normalized)

Simple oscillator Displacement/force

1

ð1 2 r2Þ þ 2jzr

ffiffiffiffiffiffiffiffiffiffi

1 2 2z2

p

1

2z

ffiffiffiffiffiffiffiffi

1 2 z2

p

Simple oscillator with velocity response Velocity/force

jr

ð1 2 r2Þ þ 2jzr

1

1

2z

Simple oscillator with acceleration response Acceleration/force

2r2

ð1 2 r2Þ þ 2jzr

1 ffiffiffiffiffiffiffiffiffiffi

1 2 2z2

p 1

2z

ffiffiffiffiffiffiffiffi

1 2 z2

p

Fluid coupling system Torque/displacement

2jzr

ð1 2 r2Þ þ 2jzr

1 1

Force transmissibility Force/force

1 þ 2jzr

ð1 2 r2Þ þ 2jzr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 8z2

p

2 1

q

2z

ffiffiffiffiffiffiffiffiffiffiffi

1 þ

1

4z2

s

(for small z)

Motion transmissibility Velocity/velocity Same Same Same

2-36 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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